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Finite motions from periodic frameworks with added symmetry
 48:1711–1729, 2011. International Journal of Solids and Structures
"... Recent work from authors across disciplines has made substantial contributions to counting rules (Maxwell type theorems) which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern, as an engineering type framework, or equivalently, as an ideal ..."
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Cited by 18 (5 self)
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Recent work from authors across disciplines has made substantial contributions to counting rules (Maxwell type theorems) which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern, as an engineering type framework, or equivalently, as an idealized molecular framework. Other work has shown that for finite frameworks, introducing symmetry modifies the previous general counts, and under some circumstances this symmetrized Maxwell type count can predict added finite flexibility in the structure. In this paper we combine these approaches to present new Maxwell type counts for the columns and rows of a modified orbit matrix for structures that have both a periodic structure and additional symmetry within the periodic cells. In a number of cases, this count for the combined group of symmetry operations demonstrates there is added finite flexibility in what would have been rigid when realized without the symmetry. Given
Frameworks, Symmetry and Rigidity
"... Symmetry equations are obtained for the rigidity matrix of a barjoint framework in R^d. These form the basis for a short proof of the FowlerGuest symmetry group generalisation of the CalladineMaxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework sys ..."
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Cited by 14 (6 self)
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Symmetry equations are obtained for the rigidity matrix of a barjoint framework in R^d. These form the basis for a short proof of the FowlerGuest symmetry group generalisation of the CalladineMaxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained pointline systems that appear in CAD, bodypin frameworks, hybrid systems of distance constrained objects and infinite barjoint frameworks. This leads to generalised forms of the FowlerGuest character formula together with counting rules in terms of counts of symmetryfixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partitionderived frameworks.
Inductive constructions for frameworks on a twodimensional fixed torus
, 2011
"... In this paper we find necessary and sufficient conditions for the minimal rigidity of graphs on the twodimensional fixed torus, T 20. We use these periodic orbit frameworks (gain graphs) as models of infinite periodic graphs, and the rigidity of the gain graphs on the torus correspond to the gene ..."
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Cited by 4 (2 self)
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In this paper we find necessary and sufficient conditions for the minimal rigidity of graphs on the twodimensional fixed torus, T 20. We use these periodic orbit frameworks (gain graphs) as models of infinite periodic graphs, and the rigidity of the gain graphs on the torus correspond to the generic rigidity of the periodic framework under forced periodicity. Here it is shown that every minimally rigid periodic orbit framework on T 20 can be constructed from smaller graphs through a series of inductive constructions. This is a periodic version of Henneberg’s theorem about finite graphs. We also describe a characterization of the generic rigidity of a twodimensional periodic framework through a consideration of the gain assignment on the corresponding periodic orbit framework. This can be viewed as a periodic analogue of Laman’s theorem about finite graphs.
The rigidity of infinite graphs
, 2013
"... A rigidity theory is developed for the Euclidean and nonEuclidean placements of countably infinite simple graphs in the normed spaces (Rd, ‖ · ‖q), for d ≥ 2 and 1 < q < ∞. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal ri ..."
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Cited by 2 (1 self)
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A rigidity theory is developed for the Euclidean and nonEuclidean placements of countably infinite simple graphs in the normed spaces (Rd, ‖ · ‖q), for d ≥ 2 and 1 < q < ∞. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in (R2, ‖ · ‖2). Also Tay’s multigraph characterisation of the rigidity of generic finite bodybar frameworks in (Rd, ‖ · ‖2) is generalised to the nonEuclidean norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit G = lim−→Gk of an inclusion tower of finite graphs G1 ⊆ G2 ⊆... for which the inclusions satisfy a relative rigidity property. For d ≥ 3 a countable graph which is rigid for generic placements in Rd may fail the stronger property of sequential rigidity, while for d = 2 the equivalence with sequential rigidity is obtained from the generalised Laman characterisations. Applications are given to the flexibility of nonEuclidean
Rigid components in fixedlattice and cone frameworks
, 2011
"... We study the fundamental algorithmic rigidity problems for generic frameworks periodic with respect to a fixed lattice or a finiteorder rotation in the plane. For fixed lattice frameworks we give an O(n²) algorithm for deciding generic rigidity and an O(n³) algorithm for computing rigid components. ..."
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Cited by 2 (1 self)
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We study the fundamental algorithmic rigidity problems for generic frameworks periodic with respect to a fixed lattice or a finiteorder rotation in the plane. For fixed lattice frameworks we give an O(n²) algorithm for deciding generic rigidity and an O(n³) algorithm for computing rigid components. If the order of rotation is part of the input, we give an O(n⁴) algorithm for deciding rigidity; in the case where the rotation’s order is 3, a more specialized algorithm solves all the fundamental algorithmic rigidity problems in O(n²) time.